Area of a Triangle - Heron's Formula

The basic formula for the area of a triangle is only helpful if you know the base and the height.

$Area of a triangle part 2 image 1$

So what do you do if you only know the three side lengths?

There are just two steps.

Step 1: Calculate half the perimeter of the triangle and call it s. $Area of a triangle part 2 image 3$

Step 2: Use the s in the following formula: $Area of a triangle part 2 image 4$

$Area of a triangle part 2 image 2$

Let's use the formula to determine the area of the triangle above.

Step 1: The perimeter of the triangle is equal to 12 + 22 + 16 = 50

Therefore, the s value is half of 50 or 25.

Step 2: Replace the s in the area formula with 25 and solve.

A = $Area of a triangle part 2 image 5$

A = $Area of a triangle part 2 image 6$

A = $Area of a triangle part 2 image 7$

A = 93.7 m²

Here are a couple more example to try.

$Area of a triangle part 2 image 8$

Example 1:

Step 1: Determine half the perimeter.

s = $Area of a triangle part 2 image 9$

Step 2: Use the s in Heron's formula.

A = $Area of a triangle part 2 image 10$

A = $Area of a triangle part 2 image 11$

A = $Area of a triangle part 2 image 12$

A = 24 units²

Now, this triangle does give the base and height. So let's compare our work to the formula you might be more familiar with using to solve.

A = $Area of a triangle part 2 image 13$

A = $Area of a triangle part 2 image 14$

A = 24 units²

Here we have the same answer as we got from using Heron's method.

Example 2:

$Area of a triangle part 2 image 15$

Step1: Determine half the perimeter.

s = $Area of a triangle part 2 image 16$

s = $Area of a triangle part 2 image 17$

s = 15.5

Step 2: Use the s in Heron's formula.

A = $Area of a triangle part 2 image 18$

A = $Area of a triangle part 2 image 19$

A = $Area of a triangle part 2 image 20$

A = 33.7 mm²

When the solution is not rational, the answer can be rounded. In this example, we rounded to the nearest tenth.

Let's Review

If you are given the three sides of a triangle, you can use the perimeter and Heron's formula to determine the area. There are just two steps.

Step 1: Determine half the perimeter.

Step 2: Use the three side lengths and the half perimeter in Heron's formula.

Some of the areas will be irrational numbers. That means that they cannot be represented as a fraction. Instead they are decimals that never end and never repeat. When this happens, the area can be rounded to whatever place value you may have been asked for.

Area of a Triangle - Heron's Formula

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